Form parameters for ship design, based upon hydrodynamic theory

V

Lab. y. Sdeepsbouwkund

Technischel Hogeschool

Ministerio de Defensa

_{D ''}

_-,

V

CANAL DE EXPERIENCIAS HIDRODINAMICAS, EL

PeXDO

Publicación núm. 78

INTERNATIONAL SYMPOSIUM

V

VON

V

V

SHIP HYDRODYNAMICS AND ENERGY SAVING

El Pardo, September 6 9, 1983

ISSHES-83j

FORM PARAMETERS FOR SHIP DESIGN,

BASED UPON HYDRODYNAM1C tHEORY

DR. JACEK S. PAWLOWSKI

Arctic Vessel and Marine Research Institute National Research Council

CANADA

I: ENERGY SAVING

Paper No. l-4

IsSl-1E$-3

Ivteiria.onaL Sqmpoum ovt Ship Hydiwdtjnam.éco ctnd Eney &tv&tg. U Patdo, Sep.tembelL 193.

PAPER 1.4

FORM -PARAMETERS FOR SHIP DESIGN,

BASED UPON HYDRODYNAÌ1IC THEORY

VR. JACEK S. PAWLOWSKT

Arctic Vessel and Marine Research

Institute

National Research Council

-CANADA

"There is no reason to helieve that

hydrodynamic theory respects the

tradi-tional, naval architecture coefficients"

Nathan K. Bales, [il.

1. INTRODTICTION

In the publication [2] the process

of ship design has been discussed as a

hierarchic deäisiori making process

and-the-concept of nonstructural model has

been introduced.

Nonstructural models

are considered to be appropriate tools

for decision making in the process of

design due to their deliberately chosen

mathematical forñ which does not match

-the complicated structure of -the

phys--ical object hut instead follows the

needs of the decision maker.

FTowever

the compatibility of a nonstructural

model with its physical object, at the

most fundamental level of the arguments

of the rnodel,-remains to he of crucial

importance.

In the present paper a network

mod-el of the hierarchic process of désian

is employed in order to elucidate the

basic dichotomy of the process, between

the descriptions of th,e design in the

function and attribute spaces, in

con-nection with the design of ship forms.

These considerations lead to the general

definition of a "dynamic" ship form

which is specified in terms, of a

des-scription in the f-unction space -of

hydrodynamic forces.

In thàt contêxt

the prohlem of designing a ship form is

understood to be the one of reducing a

dynamic ship form description to a

geo-metric description in the attribute

space.

On the basis, of these ideas the

'ñotións of necessity, sufficiency and

-equivalence of a geometric form

de-scription are formulated, which specify

he idea of .compàtibility of the aru=

'inents of a nonstructural iiodel with its

physical object.

As art example a deivation is

pres-ented, by means of a normalization of

the governing equations of strip-theory,

of a set of basic geometric form

para-meters for seakeeping studies.

These

parameters were derived for the first

time in 19Th and since then they have

been applied to the identification of

seveal noritructural --models of ship's

behaviour in waves,

[21-,

[31 ,

[41,

151

[) ,

[-7] , (A] , [91 .

n order to discuss

further general pronerties of geometric

form parameters and their meaning in the

desian of ship forms, the problem is

in-tròduced of -findina necessary variations

-,

of a geometrIc form in order to satisfy:'

a pre-assiqned correction of

hvdro-dynamic forces exerted upon the shin or

alternatively a correction of ship mo-

-tions.

It is shown by the example of

the normalized aoverninq equations of

strip-theory, how within the li-near ap-e'

proximation to the dependence of -forces

--upon form parameters a locailv,uivá-

-lent aeometric -form 'description can he

found and how "specialized" or

"rè-stricted" ship --fOrms (such as slender'.or

-symmetric- forms) -can 'be 'explained in

that context.

These id-éas, --which' are

concerned with finding -'-the

1-ocal inverse

of the geometric - form -. ship performance

causal relationship, ar-e-. further

'ill-u-tràted iñ the- Appendix' 2.

In the firìa part of the paper the

methods are discussed of assessina the

sufficiency of sets of' -geometrie form

parameters,, and approaches usually

ap-plied to the selection of the form

para-meters, other than the normalization of

the governing equations, are compared,

with examples taken from the classic

publications on seakeepina properti-es of

ships

[1,01

(111,-

11-21,

[13].

-Fssential Notation

A - a set or vector of geometric form

- the components of the vector

B -

breadth at the midship

section.$.-O

b _{local breadth at the waterplane,}

BFlLLBt1L -

lonqitudjnai metacentric

radius,

1LB1

transverse metacentric radius,

- block coefficient,

C-

the area coefficient of the midship cross-sectjon1. 0

C- waterplane área coefficient (with

respect to

BL),

D -

draught at the midship cross-Section, .xO,

- local draught,

- qeneralized hydrodynamic forces

" j1,2.-.. , Froude number - longitudinal metacentric height,

Gt11LCM

-

transverse metacentrjc height,

9 -

the acceleration of aravity,

L -

length at the waterplane,

- the radius vector in the

A,Z-

Cartesian system - the vector of linear

dis-(.Iotez.0')

placements of the ship or o the complex amplitudes of these dis-placements, depending on the context,

5 -

the reference configuration of the underwater surface of the hull S,

T - the internal stress tensor in the

fluid domain,

U. = forward speed of the ship

V - volume of displacement,

1X- longitudinal Cartesian coordinate taking valúesiLi1-lLat the fore and and aft ends of the waterplane

re-spect ively,

- the longitudinal coordinate

of the centre of bouyancy,

- the longitudinal coordinate

of the centre of flotation, transverse Cartesian coordinate positive towards port,

L - vertical Cartesian coordinateLaOat

the waterplane, positive upwards,

zcL2

the vertical coordinate of the centre of bouyancy,

zEL. the vertical coordinate of

the centre of mass, 01.- regular wave amplitude,

heave phase-lag, pitch phase-laq,

- the veçtor of anqular dis-placements of the ship or of the complex amplitudes of these displacements, in radians,

- wave length,

3 -

specific density of water, - tensor product, A - vector product, .'.- scalar product.

cross-LZ

9-f

2. SHIP FORM AND THE PROCESS OF DESIGN 2.1 A Model of the Process of Design

In reference [21 the process of

de-sign has been described as a hierarchic

decision making process. _{The development}

of approaches to preliminary ship design which are based on a network representa-tion of the process, [14], [15), suggests that network formalism can provide a deeper insight into the fundamental

prob-lems of ship design, [16].

Most appropriately, perhaps, a net-.

work can he considered as a model of the decision making process if the nodes are

thouaht of as corresponding to elementary decision making processes whereas the

di-rected branches of the network are

in-terpreted as the lines of the flow of

data. In that model the process of de-sign is realized by carrying out through the network the flow of data which tarts

from pre-assigned functional requirements

imposed upon a design and when complete1d

gives the description of the design in a

pre-specified final attribute space.

As the process progresses, according

to the hierarchic. structure of the

net-work, at every node which is reached by

the flow of data a decision is taken

as-signing to the design a description in

the attribute space of the node. The assiqnthent is guided by the requirement. to achieve an acceptable description ofj

the design in the function space of the

node. The. names of attribute and fune-1

tion spaces f011ow the nomenclature

adopted in [171.

An example is shown in Fia. 1, dis-playinq typical features of a node and at

the same time representing the starting

point for further considerations.

Size, _V Regime of motion, U. Environmental Function Space: generalized envi-ronmental forces,F FIG. i

The inflow of data to the node is

represented by the size of the ship (de-fined e.g. by the volume of displacement

V),

the regime of motion (defined e.g.. by the forward speed U.. ) and the envi-:

ronmental conditions (defined e.g. by a setH of the parameters of wave spectra), all of which have been determined at the

Attribute Space: form,

preceding (higher in hierarchy) stages

of the process. At the node the deci-sion is to be taken determiñing the hull form in such -a way that acceptable values of generàliEed environmental

forces exerted upon the ship are achieved. Instead of the forces any set of parameters characterizing ship's

re-sponses to the interaction with

environ-ment can be used for defining the

function spaÇe.

In order to càrry.out the decision

making process a model of

the-cause-effect dependence of upon

A ,

for specified

V ,

U. and H , must he

em-ployéd. The necessity to employ models at the nOdes corresponds to a

funda-mental property of decision making

processes, (2]

In the example conâidered here the -model can be expressed formally by:

Due to the dependence of upon Pt in

(1), the search for an acceptable form usually requires trials and errors approach in which a form is specified and next the resulting forces f are checked for acceptability. This

suq-gests that an explicit relation inverse

to , or in particular- a function

in-verse to , determining forms which

correspond to assumed forces, should be more suitable for the process of design.

2.2 The Description of Ship Form Based upon Generalized Fluid Forces

In order to achieve a better under--standing of the problem described above,

in terms of the fundamental properties of the process of design, let it -be no--ticed that ship forms càn he described

by means of the cOmponents of the

forces . To this end it is enough to

partition the set of forms into classes by means of the following definition:

where Pt denotes the class of forms which are such that for assumed values

of V , U. and 1 the generalized forces are equal to . Hence all such forms

are considered to be equivalent i.e. there is no need to distinquish between them and consequently they can be con-ceived as one and the same form.

This "dynamic" definition of form

displays two important properties. First, it depends on the values of the parameters V EU.. , 14 , i.e. two forms

which are the same for a given set of values of these paraméters are generally

not the same for a different set of

val-ues. Second, and this property is most important in design context, it does not

provide a description, of form in the

at-tribute space in which forms must be

specified in purely geometric terms.

Therefore, the dynamic definition of form cannot be applied directly to solving the

decision making problem. However at

-tempts can be made at reducina the

dynam-ic definition of form to a aeometrdynam-ic de-scription. Those, if successful, would

provide the most natural geometric de-scription of ship form for the process of

desïqn.- To t-his end it is possible to

utilize an analogy between geometric and

dynamic descriptions of form.

For a specified volume V the

geo-metric form of the hull is completely determined by the surface

S

of the hull. The surface S is assumed to be- regular, see (181, so that thedifferential forth:

¿5cLS

(3)

is well defined on it with the exception

of a finite number of lines which

consti-tute boundaries of regular surface ele-ments. The values óf the dIfferential

form d. describe t-he surface in terms of

elements of infiniteimal area

ctS

and normal unit vectors t4 (considered here to be directed into the fluid domain).

Usually, in applications components

ofd3

are discretized, as it' is e.g. a common practice in hydrodynamic calculations. Disadvantages of discretized form

de-scriptions in the process of design -have been discussed in (21.

Another-approach whereby the values

of d. can he reduced to a f-mite number of parameters -is tó consider intearals of ¿Son .5 . It is assumed that the surface

is the underwater surface of the hull which conforms to the usual port-starboard symmetry requirement.

r,et the following tensor moments of

the form t he considered:

M0oL Ptdaì,

(4)

with denoting the radius vector with

respect to the point of intersection of the midship cross-section with the water -plane and the plane of symmetry. If

these moments are normalized with

re-spect to LB, L2B , and

LB

respectively

it is found that their components corre-spond to the form parameters:

iicae

i

(5)

MLC+

1TCB)

where the usual N.A. notation has een used.and , L and

BT

are

normalized with respect to L . The

con-dition of constant volume of displacement -is expressed by:

and it follows that the length L. _{can be} employed to describe the size of the ship, instead of the volume V . It is

significant that the-form parameters (5) appear to be products and sums of the

quantities which are usually considered

separately as describing the ship fòrni. The generalized fluid foröes exer-ted upon the ship which correspond- to the rigid modes of ship displacement

with respekt to a reference

confiaura-tion (elastic modes will not b

discus-sed here) are expresdiscus-sed by the formula:

1(T5AT.dS'

=4,z,..,é,

(7)

with

T

denoti-na the stress tensor field in the fluid domain and the subscripts on the r.h.s. indicating the

corréspon-dinq components ôf the first vector in

the parentheses for j

l,23 and the

j-3 components of the second vector

for

j4,5é

_{. If the simple instance}

of the ship in hydrostatic equilibrium

is considered, then:

T=S9LJ

(8)

with J and z. denoting correspondinqly

the second order unit, tensor and the Cartesian coordinate measured vertically upwards from the origin. rlpon inserting (8) into

(7)

and normalizing the resul-ting expressions- with respect to the normalizing factors

_qL

for $=1,2.3 and g for

_tI5'I&

it is found that the northalized forces fulfil the relations:

-o

.(9) Ce,

-Thus, in this example a one to one trans-formation between dynamic and geometric form parameters has been found. The significance of this fact results from two properties displayed by the derived geometric form parameters:

they constitute necessary arid suffi-cient form parameters for the analy-sis of the forces involved,

they are equivalent to the dynamic description of the form.

These prOperties follow from the equa-tions (9), e.q. the set of form para-meters /L.,

D/L., 1C4, C

i

sufficient but not necessary siñce the forces 1 do not depend upon

SIL

_,C

and and the set of parameters

_B/L

DIL

c

Ce, is nOt sufficient

be-cause it does not include the parameter

-

îcuon which the force

, depends.

Resides the sufficient and necessary set. of parameters /L.

c8,

is not

eguivalent to the dynamic description of form because there is no one to one transformation between these parameters

and the forces

j

. The property of eguivalence is inmortant ¿mce e.q. if it had not been known that

and 5depend

upon the product of the parameter.s D/L and

C(in other words, that the parameters

P/Land C interact, [191) this could have been revealed only by a special

investiqation. It should be mentioned

that geometric form parametérs derived means of normalization are not unique because they depend upon the choice of normalizinq factors, which is a matter

convenience and taste.

Finally it should be observed that

the forces- j in

(7)

can be thouqhtof as moments of the differential form dSonS;, by analoqy to the relations (4). However,

in general these moments are taken with

respect to much more complicated tensor fields (r and'FAT ), than in (4), which are solutions to appropriate governing

eauations of flu-id motion. _{It is clear} that the geometric form parameters in (9)

coincide with two Of the form parameters

-in (5) due to the particular form of thé

tensor field defined by (R). It folloi.s

that necéss-ary and sufficient, in the

sense described above, sets of geometric form parameters can always he found if so

called hydrostatic forces are considered.

3. A FORM flESCRIP'PION FOR SF.AKFFPINr,

CONSIDERATIONS - -

-3.1 Normalized Forces

Seakeeping considerations are

usu-ally based upon the assumption of a

po-tential flow of ideal fluid in the fluid domain, and upon liñearization with re-spect to fluid velocities (application of linearized Rernoullj's eguation), wave

elevation, and ship displacements, e.q. (20], [21], [22]. Viscous effects are

introduced semi-empirically, e.q. _[21], (23], [241, and will not be discussed here. rinder these restrictions the

internal stress tensor field T in the

fluid is expressed by:

by

of

(10)

with representing the velocity

poten-tial.

It is convenient to consider- the

hydrostatic pressure in (10) in

coniunc-tion with the gravity force acting upon the ship. Taking the reference config-uration of- the ship as coinciding with the configuratIon of hydrostatic equi-librium, the resultants _{j ,j4,i,..}

of the forces due to the hydrostatic pressure and gravity force are found in

the well known form:

(o,

Foi

1i2_t3

(11)

j

c(_cMV,

J(cçAzo'

C&MLV

*

tXc.'kw\80j

with z0 denoting the vertical displace-ment of the origin of the reference

system fixed with the ship, repre-senting the roll angle (positive towards starboard) and denoting the pitch anqle (positive for ïmmersing bow).

The expressions (11) when

norml-ized as before yield:

í.=

=o

Zce,

(12)

ì

- (L

Ce,tCw')&,

with

£tGt1r

and _G*1L normalized with respect

tob

. Hence, it follows from

(12)

that the parameters-:

(13)

-constitute the necessary and sufficient set of geometric form parameters for the

description of lïnear hydrostatic and

weight resultants due to ship motions.

However, they are not equivalent because

the number of parameters exceeds the number of non-zero forces concerned.

It should be noticed that formally

the number of geometric parameters in

(12) and

(13)

could be reducéd by movinq the oriqin to the centre of flotation

and consequeñtly getting (in a similar manner could have been Obtained in (9)). This however would

not elimi.pate the dependce of the

fOrces and on since then any hypothetical investïqatioñ of these forces would have to begin at determin-ing 'X.Fmn order to take measurements or carry out calculations with respect to the centre of flotation *). Apart from

that in the present considerations would then appear in the limits of

inte-gration of the integrals exprssina

normalized hydrodynamic forces.

The dynamic (i.e. dependent upon

the fluid flow) components of the stress

*) _{This is a situation different from}

taking

_cç=

due to the symmetry, in

(11) and (12), which is an a priori

assumption thus limiting the reàlm of

geometric forms under consideration.

tensor field (10), when inserted into the equation Ç7), give the expressions

for hydrodynaniic forces

=,

(14)

with the inteqration taken over the reference configuration So of the

sur-face

S,

due to the linearization-. The normalized form of- the forces is

Obtained:

(15)

if the following definitions f th

nor-malized quantities are adopted:

1d

Lt

çiL.

(16)

The linearized forces-

Lj

can be pré-sented in terms of so called radiation,

diffraction, , and

Froude-T<ri.loff, , forces as- additive

com-ponents, hence: -

-+ (17)

Assuming harmonic ship motions induced by regular deep water waves, slenderness Of the hull, see Appendix 1, and plane

flow at the cross-sections, the usual

-expressions of strip-theory for thehy-drodynamic forces are obtained, e.g.

[20], 121], [25]. The forces in (17) are considered to represent complex am-plitudes of quantities varying harthonic-ally in time and the imaginary unit is introduced in the subsequen

formu-lae.

The normalized radiation forces

then take the form:

(18)

X(*

_{\)+ (jii'}

11Q.Ec

(i-f.

(19)

x( - _] i-o,. 14

1+j

-Jt&TM'

- (20)

-,.L

fat

5i

u4' k

--j

5

'IA

Inthe above expressions the dot denotes

scalar product between fOur-dimensional Cartesian vectors- and second order

ten-sors, cdenotes the normalized frequency

of enôounter: i:.I3lSI !

.fot. j=S 6 i k

k-..

(24) with: x

'JY

ott,

fø

with _{representinq the wave} frequency and ,8 denoting the direction of wave propaqatjon with respect to the ship course (positive towards port), besides

U/fji'

. The superscript ¿

indi-cates that the quantity is evaluated at the stern cross-section for a ship with transom stern and otherwise is equal to

zero. The- variable _{represents the}

normalized x coordinate corresponding to the versos it follows that

and Besides:

'

(,,re')

_) '

4'),

denote respectively the áomplex vectors of the, amplitudes of ship's linear and

angular displ1acements, with '!' normalized

with respect to L,0 .sïgnifiyina the

surge amplitude, %,sway amplitude -and yaw -amplitude (positive towards port).

-The components of the

cross-sectional added-mass and damping tensor are normalized- with respect to for

uk

; for j2- and Lc=4 or

Ç=4 and k=2_ for

,Çk4

they -are equal to zero for other choices of

indices, and therefore:

.-

knid,

fot- j1V.2.,

-(r

fa=2.

-cI

(-f

(23)

o--j4 G.ydk"2.;

j.ki

In the expressions (23) the follow-ing definitions are employed:

ctb

k.,i;

with b and

_d

denoting respectively cross-séctiona-1 breadth at the water-line and draught, and

representing radiatioh potentials for sway, heave and roll modes

correspond-ingly. Besides.:

-t

J.

frt

(25)

+41L

see Appendix 1, where

3'e<-i,i.,

_, and

áre the coordinates of the

point on the cross-sectional contour.

The normalized &iffraction forces

can he expressed as , and:

o

p&oCO4).XS

dt

f:ø.t j=2,3'

(26)

foj5i,

k1-cn5 ,k2foj;

where «Ldenotes the wave amplitude

(real) and:

-('L?;.

ils) cL

(27f)

fot

j=Z.«3

K (Z14 t',cp

The formulae for normalized Froude-Kr.ilof forces are expressible in the

form and:.

-( 2R)

Finally the normalized inertia forces are given by the formulae:

-3c( +Z*+

(30)

and

;-(-+î -«

O_

.+ke,B;-k.X (31)

+7.4j ifoi-4iS.

assuming port-starboard symmjtr of mass distribution and with

denoting appropriately defined radii of

gyration (

kcan take macmary

val-lues) normalized with respect to

L

3.2 The Parameters of Form

The formulae for the normalized forces, which have been derived in the

preceding paragraph, show that the

forces depend explicitly on the fòllow-ing geometric form parameters:

XCw C)C2CaI

₍₃₂₎

MLC

-Besides the locl normalized breadth of the waterplane j»r/Í, and normalized

profile ad./

appeàr explicitly in the

formulae.

Furthermore cross-sectional form

'parameters are involved in the inteqrals over cross-section'al contours in the ex'-pressions (23), (27) and (29). This' kind of dependence upon form is shown

partly in the formulàe (25) and is

involved implicitly in the formulae (23) and (27) due to the presence of the radiation potentials

,Z13't

The

implicit influence of form results from the radiation potentials being linked to cross-sectional form by the imperme-ability condition imposed on the flow at

the contour, which is expressed by the

normalized formulae:

r3a

1.(OlThfl3

(33)

-with denoting the normalized contour

It results from

the above discus-sion that in 'addition to the form

para-meters (32) the normalized forces depend

on the shapes of two curves

on the shapes of the 'curves

which constitute a set of

power continuum.

In order to redüce

these form descriptors to a finite num-ber of parameters the curves can be characterized by their àppropriate

mo-ments, following the approach which has.

been adopted in connection with the re-duction of the description of the hull

surface, see the equation (4). The

leading motive of this procedure is to utilize the parameters (32) to the

largest possible extent in the

descrip-tion of ship form and thereby to keep

the description limited to 'as few para-P

meters as

it is feasible. In 'this

in-stance the parameters (32) are

particu-larly'well suited to be adopted as the

fundamental form descriptors because

they appear explicitly in the equations and most of them govern the hydrostatic forces which are kngwn to have dominant

influence upon ship behaviour in waves. Thus the zero moment of the normal-ized cross-sectional contour with

re-spect to the waterplane reduces the contour to the cross-sectional area

co-efficient

c()

The 'applicability of Morrish's formula

(seeP e.g. (26], E27]) to ship-like forms

suggests that usually the first moments

of the contour shòüld not contribute considerably to the descriotion of the form, takina into account the symmetry

of the contour and that bld, parameter is

determined hyb/W0

Fence, a

sim-plifying assumption can be introduced according to which the area coefficient

c(i is sufficient for the

descrip-tion of the form of the normalized con-tour Althouqh the success of the application of Lewis form cross-sections, which are entirely determined

by the values of ,/d, and C , see e.a.

(28] , in the computation of ship motions

in waves strongly confirms this assump-tion, the assumption itself is

'indepen-dent of any such application, it can he

checked for any other sub-class of con-tours and higher order moments can be introduced if necessary.

Moments of the normalized waterline

produce the relations:

!ccc,

1Ç cLX=

BMLC

c,

h

35)

which in con-junction

with the condition:

I 5A

°i'

=1i

4Ot)'O'

.l_

--L' T0t

(34)

(3e)

and;

(37)

dtx

make it possible to approximate the waterline by means of fourth order

polsi-nomiâls in the intervals

-O7

and

on the basis of the values of the r.h.s. in the relations (35) and

(36). Such an appròximation can be modified in order to include a parallel

middle body, and is usually found to he

satisfactory, [13] , [29], for the

purpose of the investiqation of ship motions in waves.

Instead of the curve of àrea

coef-ficients CC,x('' it is more suitable

to consider the normalized curve.of areas itself which can initially be rep-resented by the expressions:

LI)

G11kc

Conttd

where the döts indicate parameters, such

as the particülars of the curves of

hiqher order moments of the cross-sec-tional contours or the length of

parai-lel middle body, which it is possible to:

derive according to the method explained

above but which have not been introduced:

here on the assumption of their less

significant influence upon the general-ized fprces.

rTsualiy when an investigation of the influence of ship form is utìdertaken

it is also possible to consider the

parameters as constants.

This leads to the reduced set of form

parameters:

Ä(

c

JCM,ZaCß

-expressed here as a vector, which, is convenient in applications, [2]. Fur-ther simplifications can be introduced by-. assuming:

---

I / *

ZtC1Cjj

(44,

on the basis of Morrish's formula, or a parabolic approximation of see e.q. [26], ánd: -. (45)

'b

C%J '.

on the basis of the paràbolic represent-ation of the waterline, analogous to

(38) with ö=O .On these assumptiôns

the forri vector _A is reduced to perhaps its most elementary size:

and any further reductions must be' sub-.

stantiated by findings of a proper study of the influence of form upon the

forces.

Resides it is possible to replace

the seventh component of the vector

, by (ç-C) Cl

. Phis corresponds to the shift of the oriqin

to but does not eliminate the influence of ce. , according to the:

discussion in the paraqraph 3.1.

The above considerations have been based on the qoverning equations of strip-theory and the question arises to what extent the form parameters are

con-CM[1-(2.Cì] -o1.. -'

<O7

(38)

cHE1-(1c--r'

Ç.o1. L

%,(107)

with:

lCp'

P

'M

and subsequently is transformed by the method of Lackenby, [30] , so that assumed values of

cand,

if required, of the length of parallel middle body are achieved.

In order to specify the profile of the underwater part of the hull it is quite enough to determine the values:

for values taken at the hase

of the stern overhänq, the end of the

keel and forefòot.' However it is

p2ssible and may be convenient to assume

d

1 for all

&37,

see e.g. [13].

By inspection, and by taking into account the relations:

it is found from (32), (35), (36) and (40) that the normalized forces acting upon the ship in waves, can be considered to be dependent on the following set of geometric parameters of ship fòrm:

ditioned by this origin. The absence of viscous effects has already been men-tioned and can be rectified by

intro-ducing additional fOrm parameters

dé-rived in a similar way from

semi-empirical formulae whióh are usually

applied in order to express forces

corresponding to phenomenà induced by

the viscosity of water. Referring to the effects of the linearization of the equations it should be mentioned that in general the nonlineàr components of hydrostatic forces depend on the de-rivatives with respect to the vertical coordinate of the moments of the water-plane, (31]. The fundamental assumption

of

trip-theory, the assumption of piane flow, is reflected in the formulae (23) and (27) by the presence of "plane" ra-diation potentials .j. This is known tO

invalidate the distribution of

hydro-dynamic forces derived from strip-theory

towards the ends of the hull. Hence if a study of the influence of qeometric form parameters upon ship behaviour in

waves is founded on a strip-theory

aiqo-rithm, an underestimation of the effects of higher order moments of the water-plane and the curve of areas or alterna-tively of the differeñtial properties of these curves at the ends, should be expected in comparison with model test results or results of computations based on three dimensiohal velocity po.en-tials.

3.3 Some General Aspects of the Descrip-tion of Form.

Ïn order to discuss further the role of the descriptïon of form in the investiqation of ship behaviour in waves let the following problem be introduced. The form of the ship isinitially de-fined by a form vector Pi

and the wave amplitude is specified as

e.g. O.O1 for some assigned values

of and of the parameters of ship mass distribution. Taking into account that the forces Ç..j

12.1.-.16,defined by the equations (12),

(17)4- (31) are represented by complex numbers, it is convenient to introduce.

real quantities ?jt..

-P.e(ftj) Fovj1i2.;.é,

-3m.(FLJ_4

F-i12-for

_t.12..

and:

,_í

e3')

ÇO1i2.i...,i Çov3.11&i...1l2.v

where. P.1!') and lri L.) denote re-spectively real and imaginary parts of complex numbers. Then the equations of the dynamic equilibrium of the ship in the seaway can be written as:

3

It is well known that these equations represent a system of linear equations with respect to the réai and imaqinary parts of the complex vector of the am-plitude of ship displacemehts

Assuminq that the value Of the main de-terminant of the system is different from zero the equilibrium amplitudes can be found whïòh determineship motions in

the recular wavé train.

It is now requested to chanqe the

form of the ship in such a manner that

the vector of equilibrium motion

ami-tudes is varied by a small vector

(d014). Assuming the differentiability of the forces with respect to the form parameters, and cönsidering linear ap-proximation, this leads to the system of

equations:

1

Ç-o.

ii.i...l2-where the real components Çkof vare defined anaioqicallv to? , and it is indicated that the comnonents of the Jacobian matrix:

1 (51)

are evaluatéd at the oriainal point of

dynamic equilibrium arid the components of the matrix are evaluated for

the original form of the ship. Thus the problem reduces to solvinq he system of linear equations (50) for

4Aa(44o...

4G,). It is ¿ssumed that the rank of the

matrix

is qreater than zero,()O

owinq to the presence of hydrostatic. force components.

If the rank of the matrix is less than 12, , it follows that the

forces do riot chanqe independently in a neighborhood of A due to changes of

fOrm. The total of 12.-p force

compo-nents depend linearly upon the others

according to the equation (50) and thereby the components of the vector cannot be specified arbitrarily.

Instead, the requested changes of the

forces P,)defined by the r.h.s. of the equations (50))must be such that the aucriented matrix of the system (50) is also of the rank p . When this

restric-tion is fulfilled the system (50) can

he reduced to the system of p linearly

independent equations: y1. 42.

2!L lca=

iL-L

taking into account the, necessary change of the indexing of the forces in

comparison with the preceding equations. It may occur that in the system

(52)n.7p, then one of the possibilities

is to 'assign arbitrary values to n-p components of the vector cj so that the system can be solved for the remain-.ing components. However, the change of

form determined in this way is not opti-mal in the sense that in general it in-volves a component which does not affect the forces. This can.be seen by corisid-ering the solution of the system (52) with

4=O,

by which p componehts of

¿A

are determined in terms of arbitrarily choseñ values of the relnaininan-p compo-nents. Hence this solution represents

an n-p

dimensional linear manifold,

the kernel of the Jacobian matrix of the systen (51), in the form space of veö-tors

A ,

in which changes of form do not affect the forces within the linear ap-proximation. It follows that only in-crements of' the vector_A along versors perpendicular to the manifold are rele-vant for the changes of forces, and new coordinates of form can be introduced, which correspond to these versors and define a locally equivalent form de-scription. In terms of such coordinates the system (52) is reduced to a system

with p unknowns and its solution gives the change of form which des not

involve any component belonging to the kernel. These ideas 'are illustrated by an example in the Appendix 2.

In order to fix nomenclature, in the instance of

p12 the form

describ-able by the vector can be named "spe-cialized" or "restricted". It has been 'shown above that the main feature of'

such a form is that the forces do not change independantly due to changes of form, e.g. by considering the equations

i and 7, of the system (49), in the

complex form:

-(c(Q&O)

(53)

for ¿ZZc*O

, it is found that

at the dynamic equilibrium point:

and therefore:

'"4

#_p*

jO

k-

. .4rL. (55)

It follows that p410 and it appears that the forms' to which the equations (49) apply are restricted, the restriction being that of the slenderness of the

hull. It is also found that the

incre-inents

d0

and o must satisfy the condition:

(56)

In order to determine the

suffi-(54) ciency of a set of form parameters at

least three procedures can be applied:

a) an'investigation which includes the descriptors (such as moments and/or

de-rivatives) of the waterplane and curve

of. areas (possibly also of hull pro-f ile), opro-f the order high enouqh to prove

that the influence upon ship performance

(preferably upon the forces) of the highest order descriptors is negligible, and therefore no descriptors of still

hiaher order should be considered as necessary (the difference between even

and odd order moments as referred to fore and aft symmetry should be

The restriction of form can also bé implied by form description and e.g. the

description by means of the vector (46) gives

_p4

and hence implies further

restriction of form in addition to slen-derness. In contrast, the description of form by means of the vector (43) does

not itself indica'te an additional re-striction of form. If however it is assumed that:

(57)

which is feasible, [2] , it follows that

9

and the form should be considered as additionally restricted The

re-striction is identifiable as due to the symmetry of the hull which eliminates

e.g. the parameter

The above discussion indicates that

the sufficiency of a set of form

para-meters should be judged according to explicit restrictions imposed upon the

class of forms which are to be

repre-sented by the corresponding vectòr of

form. Besides it follows that it can be beneficial to reduce a form description

to the set of equivalent form para-meters, which as i,t has been shown can

be accomplished locally in the neighbor-hood of the form , in order to exclude

ineffective corrections of form, not

contributing to the change of ship

mötions.

4. EXAMPLES AND COMPARISONS

In the foregoing discussion a kind

of-method has been presented accordinq to which a set of necessary form para-meters can be derived for the decision makina problem described in the

para-qraph 2.1. It has been indicated that

the sufficiency of the set can be, judaed to a certain extent on the basis of

ex-plicit restrictions imposed upon the

ship forms to be considered. Besides, if

the sufficiency of the set has been established it is possible to derive a

locally applicable set of equivalent form parameters by taking the approach

observed).

b) a comparison between predictions of

ship performance based upon

sup-posedly sufficient set of form para-meters on the one hand and on a more comprehensive form description on the

other; an acceptable result of the

comparison indicates the sufficiency

of the set of form parameters.

ó) a fitting to a set of data concerning ship performancé and acquired with

the use of a more comprehensive form description, of a nonstructural model

of regressional type which is based upon supposedly sufficient set of form parameters; an adequate fit

in-dicates the sufficiency of the set of

form parameters.

As an example'of the application of the procedure b), results of

computa-tions of ship mocomputa-tions in regular head

waves are shown in Fig. 2 and Fia. 3 against values obtained from model tests. The experimental values and the

values computed by means of an algorithm of ordinary strip-theory with two-para-meter cross-sectional representation

have been reported in [321. The other

computed values have been obtained from

a strip-theory alqorithm.ifl which the

input descriptiOn of hull form has been reduced to the vector of form (46) with the origin shifted to (CCß1o)

uCci ,CMI)Z,

(Z5)Cw)

58)

and the four point specification of the.

hull profile, these were first presented

in [291. It is seen from the

compari-son that the use of the vector (58) as the primary form descriptor has not in

any way diminished the accuracy of the strip-theory algorithm in comparison with the experimental data. It should be noticed that the vector (58) has been applied in the investigation of relative

bow motion in long-crested head seas the results of which are partly presented in

[21. These results iñdicate stronaly

that _CM values, within the usual limits of variation, do not have any

signifi-cant influence upon ship motions in the head seas, and therefore in that

in-stance it is possible to reduce the vector of form to:

PIcßIGwLrccçC

A major study of the type e)

de-scribed above, has been presented in

[10]. It covered a population of

thir-ty-four models of widely varyinq form

parameters. Startina with a

heuristic-ally selected but comprehensive set of form parameters, linear reqressional ap-proximations have been found to signifi.-cant pitch and heave double amplitudes

in long-crested head seas, which are

based on the following vector of form: (60)

"-_jL L,-

,-A

-w%'cRH

in the notation of the present paper,

however, with the definitions of the parameters referred to the lenqth

between perpendiculars. There is a

strikinq affinity between the vectors

(59) and (60). Since the vector (60)

describes points in a suhspace of the

rance of vector (59), the results

presented in (101 confirm the suffi-ciency of the vector (59) for ship form description in investigations of ship

motions in head seas. It is interest-ing to observe that through .a heuristic

approach the interactions between the

parametersD/LiCeand&, and

cç and Cw

which appear in (59) and follow from, the

normalized governing equations, could

not have been anticipated. This micht

have been one of the reasons why the

influence of %)(; upon the motions was

not found significant in (101 , in spite

of the special attention which had been

paid to 'this parameter in the study.

Another reason for the differences between the vectors (59) and (60) can result from correlations between form parameters present in the sample, e.q. a

strong correlation between CB and C.,j is

reported in [10].

It should also be noticed that 'the

models presented in F2) and (10], al-thouqh 'no direct comparison between them

is possible, show very similar trends of

the influence of form parameter.s upon ship motions. In particular the

signif-icance of C is displayed; e.a. for the

increments of 10 percent of the form parameters with respect to their averaqe

values in the sample, the models in (10]

indicate for the ship length 400ft and Beaufort number R, corresponding

incre-ments of the sianificant double ampli-tude of pitch as follows:

c,:-2.4t3,

C

1.Z36:O.l4&

,J:_O.2.2.6,&cO.003

(in degrees). These trends compare very well with those shown by the. models in

(2] for relatiye bow motion. Phis

com-patibility is cuite important since the models in (2] have been identified on the basis of a numerical experiment in which form parameters were varied

inde-pendently.

The vector of form (59) or

alterna-tively its version with shifted origin:

(61)

if it is recognized as constitutinqa

sufficient. form descriptor, does not

im-ply a restriction of ship form, apart from port-starboard symmetry, in the investiqation of ship motions in head or

following long-crested seas. In the

vectors can be reduced even further to locally equivalént form vectors. How-ever if they are applied to the investi-gàtion of ship motions in the seas comprising oblique wave components, the vectors (59) and (61) reduce by four the number of forces ?j which. can be varied independently. _{Therefore the} recogni-tion of these vectors as sufficient form

descriptors then corresponds to an addi-tional restriction of ship form.

So far the fbrm parameters have

been discussed with no reference _being

made to the regime of motion and

envi-rorimenta]. conditions. _{However it has}

been pointed out in the paragraph _2.2

that thedynamic definition of form

involves the dependence of ship form upon values of the parameters describinq the aforementioned data. _{Usually the} forward speed, or one of its normalized analogues, is Öonsidered as a parameter in seakeeping studies, together with Beaufort number, significant wave height and a characteristic period of the wave spectrum as possible seaway descriptors.

The normalized equations of strip-theory

presented in the paragraph 3.1 show that a significant interaction between Froude number and geometric form parameters can occur, see e.q. the equations (19), (20)

and (23).

An example of nonstructural models

in which that kind of interaction appears explicitly, leading to the in-troduction of a form-speed parameter which can be thought of as a "dynamic form" parameter, is provided by the general models, [2] , of added resistance

in lonq-crested head waves, that have been presented in [7]. _The identi-fication of these models has been based on a version of the vector of form (61) in which, however, the components _{a and}

, are referred to the length between perpendiculars rather than the length at

the waterplane. The models are ex-pressed by the formula:

4(O(,)k

*Fr F6(°,

(62)

where O..,. represent the components of _the

vector (61) and:

(1ok" R

flW

1't. 055

_H'y

with H4,'denotinq the significant wave height and _signifyina respec-tively zero and first order moments of

the wave spectrum. The parameter o(.. is

the dynamic form parameter:

Besides, oCi

, again referred to the

].enqth between perpendiculars. _hese

models have been identified forT1e(1..'j) and Fp.EKO.4OQ, O.3.2.6' , with the

para-meter _{lE defined by the rélation:}

-where T'('

. The functions

of the formula (62) are shown _{in Fig. 4}

for _{i. and}

_Z.o

_{, and it is}

seen that the parameter o& _dominates

other parameters in the models, _{it can}

also be observed that c _{is not as}

important a parameter of form in this instance as it appears to be _{for ship} motions, [2), _{(101. This is in}

qualita-tive agreement with the results pres-ented in (101 and [11] in which regres-sional models of power and thrust

increase in lona-crested head waves have riot shown a siqnigicant influence _of

Apart from heuristic selections _of geometric form parameters for the inves-tiqation of ship Performance _{two other}

approaches are usually employed. _{One is}

to hase the investiqation on _a

system-atic series of ship forms. In fact this indirectly is also a heuristic _selection

since the form parameters of _{a series}

are as a rule selected heuristically.

However, owing to methods being applied to the qeneration of series forms these

forms usually are describable _{by a small}

number of form parameters whereas _other parameters, which should be considered independently for a ship form outside

the series, are strongly correlated with' or functionally dependent upon the basic:

ones. An example in seakeeping is provided by the nonstructural models _of

ship behaviour in lonq-òreste head

waves, that have been presented in [12]. These models have been identified _{on the} basis of an extended Series 60 with the

description of ship form reduced to the

vector:

-

IL,P

-in the nomenclature of the Present paper, however, with al]. parameters referred to the lenqth between perpen-diculars. _{Although this vector of form}

does not itself imply an additional restriction of ship form for a study of ship behaviour in lonq-crested head waves, it does not include the

para-meters Cwand

c.which occur in the

form vectors (59) and (61). However,. ás

it is explained in [12],

_Carid

_{'cc are} functionally dependent on CB in the

series.

In general, a strong correlation or

functional dependence, within a series or sample of forms, between most impor-tant form parameters may lead to results

which are difficult to extrapolate out-side the series or sample and _cumbersome

to interpret. For instance a positive

correlation between C

and c, ,

the parameters the variatiOn of .which would

most often produce opposite effects upon

ship motions, [23 , [10] , can result in a

weak variation of seakeepina performance within the series or sample, and may show reversed trends of the influence of form variation in different ranqes of

c (C if the results are interpreted

on the basis of

The other process of the selection of ship form parameters is founded on an

adequate mathematical description of the

form. An example of such a process is

provided in [131. The models of ship motions in long-crested head seas

pre-sented in [13] have been based on a form

description sufficient for the

approxi-mation of the waterplane and the curve

of local sectiOn area coefficients,C(iÌ,

by means of fourth and second order polynomials respectively with a parallel

middle body taken into account. The profile was assumed to he rectanqülar.

The corresponding vector of form can be

expressed as:

(67)

with the coefficients referred to the length between perpendiculars, CW and

denoting respectively waterplane

coefficients forward and aft midship,

and C and _CA denotiriq the limit

values of the sectional area coefficient

at the forward and aft perpendiculars correspondinqly. It follows that the

vector of form (67) constitutes a

sufficient form descriptor in the sense in which e.g. the vector (46) does.

In general, form parameters

de-rivable from an adequate mathematical

description of the hull form are suffi-cient and make possible an investigation

of the influence of the variation of

form parameters taken independently.

However, owinq to the way in which such

parameters are derived, with no

refer-ence to the governing equations, they do

not appear explicitly in the equations and hence they cannot indicate sòme nec-essary interactions or be easily iden-tified with particular physical effects (compare (46) with (67)).

5. CONCLUSION

The network model of ship design as

a hierarchic decision making process

proves to constitute a useful tool for a better understanding of the role of

fluid dynamics in ship design. Within this model the proper process of design-ing a ship form is recoqnized as takdesign-ing

place in the function space of

general-ized fluid forces affecting the ship *).

It can be said that a successful desian

imposes right proportions upon the forces (e.a. small

resistance/displaçe-ment ratio). The solutioñ in the func-tion space must be translated into -a geometric description of form in the -at

tribute space.. It has been shown above

how relätionS imposed upon the forces correspond to restrictions of the

geom-etry. This approach, more balanced than

the usual "ship form leadina to ship performance" understanding of the prob-lem, is fully expressed in the concept of the equivalent qeometric form

para-meters which remaih in one- to one

rela-tion with the independent force

com-ponents.

The translation into the attrIbu-te space requires the exisattrIbu-tence of a

suitable geometric form description which should he derived from the

egua-tions determining the fluid forces. It

has been demonstrated how a sufficient aeometric form vector can he derived

from the governing equations of strip-theory by means of the normalization of

eauations. The form vectors (59) and

(1) can be considered as sufficient

qeotnetric form des'criptors for the

in-vestiaation of ship behaviour in

long-crested head or föllowinq seas.

How-ever, more caution should he exercised if oblique wave components are to be taken into account since then these vectors do imply a restriction of form

beyond symmetry and slenderness. The

success of the normalization technique

in derivino the vectors (59) and (61)

depends stronaly on the dominant role of the hydrostatic forces in the governino

equations. For problems in which fluid flow effects are dominant stronaer

math-ematical tools may have to be applied in order to derive so clearly defined

aeo-metric form parameters.

ACTNÖWLFflr.VMFNm

The author would like to express his indebtedness to Dr. Janusz Stasiak

of The Technical Tiniversity of cdansk,

whose ingenious efforts and suòcess in

identifyiria several nonstructural models

of ship behaviour in waves have inspired the author to continue his interest in links between the geometry and fluid

dynamics of ship forms.

*) Here the line of thinking developed in (17] is followed.

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December 1969, pp. 284, 298.

(29]Pawlowski, J.S., and Kwiek, K., "The

Computation of Ship Motions in

Lonqitudinal Waves. Based Upon the

Main particulars of The Underwater

Form",. IV-th Symposium of Ship Hydrodynamics, Gdansk 1977, in Polish.

[30]Laókenby, H.! "Oñ the Systematic

Geometrical Variation of Ship Foths"

RINA Trans. 1950, pp. 289, 316.

[31]Pawlowski, J.S.,

"Basic Relatioñs of

Strip-Theory, Part II, Elements Of

the Equations of Motion", Delft

University of Technology, Ship Hydromechanics Laboratory, Rep. No

558, 1982.

[32]Gerritsma, J., Beukelman,

W.,"Com-parison of Calculated and Measured Heaving and Pitchinq Motions ôf A

Series 60, cOiO Ship Model in

Regular Longitudinal Waves", Pro -ceedings of the 11-th ITTC, Tokyo 1966, pp. 436, 442..

APPENDIX 1

Some Details of the Derivation of the

Normalized Eauations of Strip-Thèory *) It is assumed that the surface of

the hull can be parametrized piecewise

in one or both of the following ways:

(A1.1)

with

K-4

see Fig. 5.

*) With the exception of small chanqes of notation the formulae in this paper are compatible with the

formulae in [25].

By definition the differentiàl form ot'S is expressible as:

(A1.2')

with thé upper or lower term in the

pa-rentheses applicable accordinq to the. parametrization employed. Neq,lecting

the partial deri,vat:ives of ₉ and z with

respect to X as small owina to the as-sumption of the slenderness of the

hull,the form

dS

is reduced to:

ddCd.

. cL.x..

(A1.3,)

(o

This for

dCadftct

qives:

-1

D

'

ndcdC=

i- -t\.cLt

(A1.4)

'acj-

Ii'

4.o.

= 13

where onl-r the upper expression in the parentheses, is used in the equation (25) for the sake of conciseness.

Similarly:-(rcL

(Al .5) gives for

dc6d:

tcU'

bd, .W Bb

(A1.6)

rr

:3d

and agaIn only the upper expression of

the r.h..s. is used in the equation (25).

APPFNDIX 2

An Example of the Analysis of the

Influence f Form Parametèrst.ipon the Conflauration of Equilibrium..

In this Appendix an example is presented in order to illustrate some of

the considerations of the paraqraph 3.3.

Let the ship he assumed to remain in hydrostatic equilibrium under the action of generalized extetnal forces

,'which do not depend on ship form and ship displacements with respect to the reference configuration. On the.

basis of the equations (12) the vector

of fôrm is defined as:

and the equations of equilibrium, which

correspond to the equations (49) in the

more general considerations, become:

-az0 *ae

-t;.=O

(A2.2)

0-(+03

+5=O

The problem of finding the linéar

ap-proximatiOn to the change of ship fOrm which results in the increment_of the

vector of ship displacements d

_(d01d.p1

dO) leads to the system of équations:

-;

e

O

d1

G O o , ? dQa.

= 44

(l2.3) O

_dQ,

5 where:

a.

a

O %O..4% O d -O.

O i

+0

d.o

The Jacobian matrix of the system of

equations (A2.3) is of the rank

e.g.

if however

P'2..

Let(p=O and:

2.O(?O1&O,

(NO

the rank

(A2.4)

(A2. 5)

is' reduced to

(A2.6)

o,

it follows that

dO

_{, which implies} according to (A2.4) that

dq=O.

Henced

cannot be specified arbitrarily. _Since values of da.4 becôme irrelevant, for

further considerations it is convenient to diminish the dimension of the forni space by one, introducing the fOrm vector:

(A2.7)

and reduce the system of equations (A2.3) to:

-z0

O

da.,

d-&d,_

I(A2.8)

(re,-e

4Ç-(-zo'daJ,

wher.e do.,, can be chosen arbitraril e.q.

Thereby a solution for is

obtained:

' (A2.9)

However, assumina

_d13

in

(A2.8), it is found that the kernel _K. of the Jacobian matrix of the system (A2..8) is given by the parametric equâtion:

o.4

-in the equations (A2.A),provides the

system Of. eauations from which the opti-mal change of form can be determined.

*)

_{Assuming 10O6.}

(A2. 1O)

with representinq the parameter, and

in general:

('A2. 11)

which ieañs that the increment dKof the form vector comprises a component

not contributing to the chanqe. of the configuration of equilibrium. The in-crements

_dA

_{of the forni vector, which} do not inìvolve. süch a component _{can be} expressed4e.g. as:

f_

+e

(A2.12)

. ,

-where C1Q..I

and drepresent cordinates

on mutually orthoagnal axes in the plane perpendicular to

K.

_{Therefore the}

Substitution:

da.,

1

-1

e

..

ac¼-s

do

(A2. 13)

t-5 0.5

o

05

o

t.o

o V o q

.0

e

q

q

-

_'2.0

_'X/L

90

z

to

o

_.$

w

i ZoI

L.

-o

90

g

o

-o

o

ezó

.0 C

q

V o C .0 V X V

.5

0

_'IL

X

05

-0

_Q

q

o C

q

V

o."

_1.0

_.5

_'Z.o

'NL

t-0

o

t Zo

i 2.o

o

"A.

.5

X Q

I Zol

o'

I Loi

o'

O5 0

Lo

Fu :0.

ç, X X o V

e

o

o

-o

'2.0

_.o

_2.o

SAIL

90

o

-90

o ç, o V e X X X

'io

%/

_'2.o

RIG. 'Z

COMP4ISOt.4 O t-IEAV

Sø»..5E opróf,

X

EX'IMNr C

Z ]

V

CoMLa7A1IÓN CZ J

O

CoMør1-ArI.J

C 3 9 X s,

1.5

0.5

V - V V w

e'

V

o

15

-o.5

o

oi5

e

2

w X. V w

o

O

Eèo

Ô

-90

Q

e

o

ç, V V X C -

.s

o

'ÁIj

1.0

lL

90

o

V

O

-90

e

10

.à'(t

.

f.0

2.o

'/L

P.1G. 3

0.5

o

C C. X w f .5

i.0

e,'

1.5 1.0

o.

15

iO

FñtO;25

x V

q

X C

o

9

V, o V g % Q

_90

;9 f,

5

2:,

_1VL

COMPASO CP

TC4

2PJS OPTÒ.S

2o

''/L

V

o

1.0

X.

x

_vr

9

o

V. V' o

o

V

EXP

tMaj-

C 'Z J

CCML.rrriàM

C _J ÓMPU1ATLOU

C 29 J

o

2

f

ç3 o

-'z

-b

L

ro

'4

-2

-3

2o

16

lo

O

s

'2b L J I - i --. '

2.

2'S -- 26 - .

I-05

TO

LO4

i I

.11

I J

-5 -4-3 -z

i

o

i

'2

-loo.5

-.AC.D

ML)LA